The expression `[(1+sin(pi/8)+icos(pi/8))/(1+sin(pi/8)-icos(pi/8))]^8` is equal is

Options

A. 1
B. `-1`
C. i
D. `-i`

Answer

B

Explanation

Let `sin""(pi)/(8)+icos""(pi)/(8)=z`
`rArr [(1+sin""(pi)/(8)+icos""(pi)/(8))/(1+sin""(pi)/(8)-icos""(pi)/(8))]^(8)`
`=((1+z)/(1+(1)/(z)))^(8)=z^(8)`
`=(sin""(pi)/(8)+icos""(pi)/(8))^(8)`
`=(cos""((pi)/(2)-(pi)/8)+isin""((pi)/(2)-(pi)/(8)))^(8)`
`=(cos""(3pi)/(8)+isin""(3pi)/(8))^(8)=cos3pi=-1`